THERMAL PROPERTIES

Thermal property is the response of a material to the application of heat. As solids absorbs energy in the form of heat, its temperature rises and its dimensions increase. The energy may be transported to cooler regions of the specimen if temperature gradients exist, and ultimately, the specimen may melt. Heat capacity, thermal expansion, and thermal conductivity are properties that are often critical in the practical utilization of solids.

HEAT CAPACITY

It is a property that is indicative of a materials ability to absorb heat from the external surroundings; it represents the amount of energy required to produce a unit temperature rise. In mathematical terms, the heat capacity is expressed as: where is the energy required to produce a temperature change. Heat capacity is specified per mole of material (, or ).

Heat capacity: ratio of energy change (energy gained or lost) and the resulting temperature change.

Specific heat (often denoted by a lowercase ) is sometimes used; this represents the heat capacity per unit mass and has various units (, , ).

(Schematic representation of the generation of lattice waves in a crystal by means of atomic vibrations.)There are in fact two ways in which this property may be measured, according to the environmental conditions accompanying the transfer of heat: One is the heat capacity while maintaining the specimen volume constant, which is denoted ; and the other is for constant external pressure, which is denoted .The magnitude of is almost always greater than ; however, this difference is very slight for most solid materials at room temperature and below.

Vibrational Heat Capacity

In most solids the principal mode of thermal energy assimilation is by the increase in vibrational energy of the atoms. Again, atoms in solid materials are constantly vibrating at very high frequencies and with relatively small amplitudes. Rather than being independent of one another, the vibrations of adjacent atoms are coupled by virtue of the atomic bonding. These vibrations are coordinated in such a way that traveling lattice waves are produced, a phenomenon represented in figure.

THERMAL EXPANSION

Most solid materials expand upon heating and contract when cooled.

where and represent, respectively, initial and final lengths with the temperature change from to . The parameter is called the linear coefficient of thermal expansion; it is a material property that is indicative of the extent to which a material expands upon heating and has units of reciprocal temperature [ or]. Of course, heating or cooling affects all the dimensions of a body, with a resultant change in volume. Volume changes with temperature may be computed from

where and are the volume change and the original volume, respectively, and symbolizes the volume coefficient of thermal expansion.

Tabulation of the Thermal Properties for a Variety of Materials

THERMAL CONDUCTIVITY

Thermal conduction is the phenomenon by which heat is transported from high- to low-temperature regions of a substance. The property that characterizes the ability of a material to transfer heat is the thermal conductivity.

It is best defined in terms of the expression

where denotes the heat flux, or heat flow, per unit time per unit area (area being taken as that perpendicular to the flow direction), is the thermal conductivity, and is the temperature gradient through the conducting medium. Units of and are and , respectively. This equation is valid only for steady-state heat flow that is, for situations in which the heat flux does not change with time. Also, the minus sign in the expression indicates that the direction of heat flow is from hot to cold, or down the temperature gradient.

Mechanisms of Heat Conduction

Heat is transported in solid materials by both lattice vibration waves (phonons) and free electrons.

Thermal conductivity is associated with both mechanisms, & the total conductivity is sum of the two contributions. where and represent the lattice vibration and electron thermal conductivities, respectively; usually one or the other predominates. The thermal energy associated with phonons or lattice waves is transported in the direction of their motion. The contribution results from a net movement of phonons from high-to-low temperature regions of a body across which a temperature gradient exists. Free or conducting electrons participate in electronic thermal conduction. Again in kinetic energy is imparted to the free electrons in a hot region of the specimen. They then migrate to colder areas, where some of this kinetic energy is transferred to the atoms themselves (as vibrational energy) as a consequence of collisions with phonons or other imperfections in the crystal. The relative contribution of to the total thermal conductivity increases with increasing free electron concentrations, because more electrons are available to participate in this heat transference process.

(Thermal conductivity versus composition for copperzinc alloys.)Metals

(Dependence of thermal conductivity on temperature for several ceramic materials.)In high-purity metals, the electron mechanism of heat transport is much more efficient than the phonon contribution because electrons are not as easily scattered as phonons and have higher velocities. Furthermore, metals are extremely good conductors of heat because relatively large numbers of free electrons exist that participate in thermal conduction. The thermal conductivities of several of the common metals are given in table above; values generally range between about 20 and 400 W/m.K. Because free electrons are responsible for both electrical and thermal conduction in pure metals, theoretical treatments suggest that the two conductivities should be related. Alloying metals with impurities results in a reduction in the thermal conductivity, for the same reason that the electrical conductivity is diminished; namely, the impurity atoms, especially if in solid solution, act as scattering centers, lowering the efficiency of electron motion.

Ceramics

Nonmetallic materials are thermal insulators inasmuch as they lack large numbers of free electrons. Thus the phonons are primarily responsible for thermal conduction: ke is much smaller than kl. Again, the phonons are not as effective as free electrons in the transport of heat energy as a result of the very efficient phonon scattering by lattice imperfections. Thermal conductivity values for a number of ceramic materials are contained in table; room-temperature thermal conductivities range between approximately 2 and 50 . Glass and other amorphous ceramics have lower conductivities than crystalline ceramics, because the phonon scattering is much more effective when the atomic structure is highly disordered and irregular. The scattering of lattice vibrations becomes more pronounced with rising temperature; hence, the thermal conductivity of most ceramic materials normally diminishes with increasing temperature, at least at relatively low temperatures (figure at right).

As Figure indicates, the conductivity begins to increase at higher temperatures, which is due to radiant heat transfer: significant quantities of infrared radiant heat may be transported through a transparent ceramic material. The efficiency of this process increases with temperature. Porosity in ceramic materials may have a dramatic influence on thermal conductivity; increasing the pore volume will, under most circumstances, result in a reduction of the thermal conductivity. In fact, many ceramics that are used for thermal insulation are porous. Heat transfer across pores is ordinarily slow and inefficient. Internal pores normally contain still air, which has an extremely low thermal conductivity approximately 0.02 W/m.K. Furthermore, gaseous convection within the pores is also comparatively ineffective.

Polymers

As noted in table, thermal conductivities for most polymers are on the order of 0.3 W/mK. For these materials, energy transfer is accomplished by the vibration and rotation of the chain molecules. The magnitude of the thermal conductivity depends on the degree of crystallinity; a polymer with a highly crystalline and ordered structure will have a greater conductivity than the equivalent amorphous material. This is due to the more effective coordinated vibration of the molecular chains for the crystalline state.

Polymers are often used as thermal insulators because of their low thermal conductivities. As with ceramics, their insulative properties may be further enhanced by the introduction of small pores, which are ordinarily introduced by foaming. Foamed polystyrene is commonly used for drinking cups and insulating chests.

THERMAL STRESSES

Thermal stresses are stresses induced in a body as a result of changes in temperature. An understanding of the origins and nature of thermal stresses is important because these stresses can lead to fracture or undesirable plastic deformation.

Stresses Resulting from Restrained Thermal Expansion and Contraction

Let us first consider a homogeneous and isotropic solid rod that is heated or cooled uniformly; that is, no temperature gradients are imposed. For free expansion or contraction, the rod will be stress free. If, however, axial motion of the rod is restrained by rigid end supports, thermal stresses will be introduced. The magnitude of the

stress resulting from a temperature change from to is where is the modulus of elasticity and is the linear coefficient of thermal expansion. Upon heating , the stress is compressive , because rod expansion has been constrained. Of course, if the rod specimen is cooled , a tensile stress will be imposed . Also, the stress in equation is the same as the stress that would be required to elastically compress (or elongate) the rod specimen back to its original length after it had been allowed to freely expand (or contract) with the temperature change.

Stresses Resulting from Temperature Gradients

When a solid body is heated or cooled, the internal temperature distribution will depend on its size and shape, the thermal conductivity of the material, and the rate of temperature change. Thermal stresses may be established as a result of temperature gradients across a body, which are frequently caused by rapid heating or cooling, in that the outside changes temperature more rapidly than the interior; differential dimensional changes restrain the free expansion or contraction of adjacent volume elements within the piece. For example, upon heating, the exterior of a specimen is hotter and, therefore, will have expanded more than the interior regions. Hence, compressive surface stresses are induced and are balanced by tensile interior stresses. The interior-exterior stress conditions are reversed for rapid cooling such that the surface is put into a state of tension.

Thermal Shock of Brittle Materials

For ductile metals and polymers, alleviation of thermally induced stresses may be accomplished by plastic deformation. However, the non-ductility of most ceramics enhances the possibility of brittle fracture from these stresses. Rapid cooling of a brittle body is more likely to inflict such thermal shock than heating, because the induced surface stresses are tensile. Crack formation and propagation from surface flaws are more probable when an imposed stress is tensile. The capacity of a material to withstand this kind of failure is termed its thermal shock resistance. For a ceramic body that is rapidly cooled, the resistance to thermal shock depends not only on the magnitude of the temperature change, but also on the mechanical and thermal properties of the material. The thermal shock resistance is best for ceramics that have high fracture strengths f and high thermal conductivities, as well as low moduli of elasticity and low coefficients of thermal expansion. The resistance of many materials to this type of failure may be approximated by a thermal shock resistance parameter TSR:

Thermal shock may be prevented by altering the external conditions to the degree that cooling or heating rates are reduced and temperature gradients across a body are minimized. Modification of the thermal and/or mechanical characteristics in equation may also enhance the thermal shock resistance of a material. Of these parameters, the coefficient of thermal expansion is probably most easily changed and controlled. For example, common sodalime glasses, which have an l of approximately , are particularly susceptible to thermal shock, as anyone who has baked can probably attest. Reducing the and contents while at the same time adding in sufficient quantities to form borosilicate (or Pyrex) glass will reduce the coefficient of expansion to about ; this material is entirely suitable for kitchen oven heating and cooling cycles. The introduction of some relatively large pores or a ductile second phase may also improve the thermal shock characteristics of a material; both impede the propagation of thermally induced cracks.

It is often necessary to remove thermal stresses in ceramic materials as a means of improving their mechanical strengths and optical characteristics. This may be accomplished by an annealing heat treatment.